76 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA where the xt are points of E, and r > 0. Suppose that H satisfies the condition (12.15.7.1). Choose e > 0 such that 2sc < %r. For each index /', there exists an an(i) such that xt - an(i) e eV. By virtue of (12.15.7.1), it follows that for each x' e H we have |<X- - an(i) ,x'-x'0y\£ 2&c < ir, and the relations |<flrt(0' x' — x'0y\ < %r therefore imply the inequalities (12.15.7.2). Since the weak topology is Hausdorff, we have shown that H is metrizable (12.4.6). More precisely, the above argument shows that the map- ping ;K'!-»«#„, -x'>)n£o °f H into the product space KN (where K = R or C) is a homeomorphism of H onto a subspace L of KN. So it remains to be shown that L is compact. In the first place, the projections of L are boundedsubsets of K (and there- fore relatively compact ((3.17.6) and (3.20.16))): for each index n there exists ln > 0 such that lnan e V and therefore |<aw, x'>| ^ c/ln for all xf e H. By virtue of (12.5.9), (12.5.4), and (3.17.3), it remains to be shown that L is closed in KN. Now if (x'm) is a sequence of points in H such that for each n the sequence «an, x'my)m^ converges in K, then it follows from (7.5.5) that the sequence (x'm) converges simply in V. Since for each z e E there exists a scalar y such that yz e V (so that <z, *4> = y"1^, x'my)9 the sequence (x'm) converges simply in E: its limit yr is a linear mapping, by the principle of extension of identities, and is continuous by (7.5.5), and is therefore a point of H. Q.E.D. We recall that if E is a normed space, its dual E' = <&(E; K) is a Banach space with respect to the norm \\x'\\ = sup |<x, x'y\ (5.7.3); since KJC, xry\ ^ \\x\\ - \\x'\\> the topology defined by this norm (sometimes called the strong topology on E') is finer than the weak topology on E'. (12.15.8) Let E be a normed space. The mapping x'\~* \\x'\\ ofE' into R is lower semicontinuous with respect to the weak topology. For it is the upper envelope of the weakly continuous mappings x'*-»!<#, x'y\, where x runs through the ball \\x\\ ^ 1 in E. Hence the result follows from (12.7.7). c£) is a sequence in E' which converges weakly to a', then by (12.7.13) we have (12.15.8.1) 'fa + u2) = \ + * u2 , \Xu) =A'*u